Whenever I get worried that Bobby Jindal has a chance of winning the presidency, I try to be hopeful that the Dems can hang this stuff around his neck. .
That pbs piece appears to be lifted more or less wholesale from Mother Jones. For shame.
I'm having one of those weeks where it's hard to remember that a majority of people did actually vote for Obama and Democrats. I feel like I'm crushed by throngs of rightwing assholes over here lately.
The first rule of theodicity is that it is easy to explain the suffering of other people. Especially the suffering that your people are responsible for.
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Gross?
The Delights of Disgust, or Martha Nussbaum is an idiot, which I already knew + Oral Sex! ...Chronicle of Higher Education, thanks Rittholz. The author says that a publisher impregnated one of the best scientific books on disgust with a disgusting odor. I find that hard to believe.
The particular form of disgust in this post, which might be analogized to a reaction to someone wearing Nazi paraphenalia, were I to break blog etiquette, is very interesting. Disgust at someone else's moral judgement is a little akin to Nussbaum's argument (about teh gay sex). Nussbaum goes very wrong in then saying all disgust is disgusting, which besides being self-refuting, ...never mind
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3: One of the best parts of the Inauguration for me was remembering that Barry won a real, live majority, even though (gasp - he had to say it out loud) his middle name is Hussein. Maybe We the People actually know what we're doing.
"Impregnated" is accurate, but I still might better have used "infused"
At the faculty/staff lunch this week, I was in line with a bunch of chortling assholes about the fact that "right" has two meanings! It means correct and tea party horrific bullshit! Right is right! There is no fucking way I'm going to engage with any of that, yet it makes my blood boil to just quietly get my soup and sandwich and sit down.
I like to say I'm infused with fetus. (I opted not to make the grosser infused-with-semen joke so all of you can just keep that to yourself.)
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Well, if we are going all moist on Obama, here's Matt Taibbi on Mary Jo White, soon to be head of the SEC. Crooked to a hilarious degree, and way cozy with Republicans.
Enough inspiration from the web, back I go to Marxist theories of the state (great author name, Clyde Barrow) and bourgeois revolutions (Deutscher winners are the best)
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At the faculty/staff lunch this week, I was in line with a bunch of chortling assholes about the fact that "right" has two meanings! It means correct and tea party horrific bullshit! Right is right! There is no fucking way I'm going to engage with any of that, yet it makes my blood boil to just quietly get my soup and sandwich and sit down.
Just be thankful that English, unlike several other languages (Italian, French, German), does not have the parallel phenomenon where the word for "left" carries unmistakably negative connotations.
Mary Jo White is a lovely person and if he's such a great reporter Taibbi should know when a firm name includes an ampersand and "LLP" at the end.
12: I enjoy referring to people who use pens the same way I do as the fellowship of the sinister.
12: "Sinistra! Sinistra! Ah, bufone!"
The Mother Jones article cited by Knecht (and the pbs article, meanie!) also has this one:
11. Abstract algebra is too dang complicated: "Unlike the 'modern math' theorists, who believe that mathematics is a creation of man and thus arbitrary and relative, A Beka Book teaches that the laws of mathematics are a creation of God and thus absolute...A Beka Book provides attractive, legible, and workable traditional mathematics texts that are not burdened with modern theories such as set theory."--ABeka.com
16
As Blume says (VTSOOBC!) what did set theory ever do to them? No, really, I'm curious. Anybody know?
The reasonable form of this argument is that by trying to cover too much you end up neglecting the more important stuff. Set theory is just as an example of the less important stuff.
16 - I read some speculation about it when that Mother Jones article was circulating, but she couldn't come up with a satisfactory answer.
I feel like I'm crushed by throngs of rightwing assholes over here lately.
The last couple days everyone here has been crushed by throngs of well-scrubbed midwestern church-group teenagers wearing "LIFE" hats and sucking up all the air, apparently in town for some Santorum-based post-Roe-anniversary anti-choice festivities. So depressing. I was thinking, don't 15-year-olds from Wisconsin and Kansas* have better things to do, like go to school? But I guess that wouldn't necessarily be a huge improvement.
*Lest I appear to be making unwarranted assumptions about where these throngs came from, they came equipped with nametags displaying their geographical details.
I wonder if it has to do with the incompleteness theorem; I could see the idea that it was impossible to construct a perfectly self-consistent axiomatic basis for mathematics as pretty darned troubling for fundies.
Wasn't "new math" a fairly mainstream movement to start teaching set theory early, because, of course, once you've mastered set theory understanding arithmetic becomes trivial.
Oh, the briefly trendy attempt to inject set theory to the elementary curriculum was a bad idea, as a matter of pedagogy. The motivation seemed to be that since set theory is *foundation* to mathematics (in some sense) you can help little kids get a deep understanding of math if you introduce them to those foundations. It was all based on a confusion between logical foundation and psychological mechanisms of understanding.
But this isn't a religious objection, and no one teaches set theory to 4th graders anymore anyway. A lot of fundamentalist pedagogy seems to be about continuing to fight the battles of decades past.
I have no idea what "new math" was or whether I learned it, but Erma Bombeck certainly made fun of it a lot.
In all seriously, 24 is wrong. But I have to go grocery shopping instead of explaining what little I know about the California Math Wars and all that.
I think I did learn it, and it was fine as far as that went.
26: That's where I've gotten most of my information on the subject.
25: That confusion was pretty common, albeit in more minor forms, in the high school texts I remember, emphasizing axioms over motivating questions and problems.
As far as I can remember the entire content of any set theory I ever learned was notation: that you draw a little U for union and an upside-down one for intersection, and put things in {} if they're a set. It was all completely trivial, right? No one was ever teaching the Zermelo-Fraenkel axioms to elementary school children.
I remember in third grade we had to go through one door if we chose to accept the axoim of choice and a different one if we didn't. I don't know what ever happened to the other kids.
It's still possible that ABeka has the same misconception I do, along with being outdated.
They all had to become construction workers.
I wonder if it has to do with the incompleteness theorem; I could see the idea that it was impossible to construct a perfectly self-consistent axiomatic basis for mathematics as pretty darned troubling for fundies.
Just think how shocking the incommensurability of the diagonal of a square was to the Pythagoreans.
35: it's also possible that ABeka is under no misconception but is marketing its books to people whom they think they can scare/shock by implying that other math texts do teach (a) set theory (b) constructivist philosophies of math.
It's well known that most fundamentalist parochial schools stand firmly with Kurt Gödel in affirming mathematical Platonism.
From that link: Remember how the modern idea of set theory really isn't all that modern? That's because I'm pretty sure A Beka doesn't mean "modern" as in "recent", but "modern" as in "modernist".
No! They mean "modern" as in "modernity"!
That post fails surprisingly comprehensively to increase one's understanding of what's going on.
How about some definitive advice based on the collective wisdom so often displayed here?
1) Cedars called, wanted me to come in to donate a unit of my precious bodily fluid 'cause they're running low and mine is apparently wonderful.
2) I agreed, set up the appt. for Monday.
3) The local news is showing ERs around the area packed with slobs spewing saliva and snot in symphonies of spasmodic sneezes and coughs.
Should I cancel the appointment, keep it, or just throw some random useless fat fuck onto the tracks and hope for good result?
That Cedars thing is a phishing scheme run by vampires, dude. /watched half an episode of True Blood last week and now worries
I always wondered about this crazy "new math" that Charlie Brown complained about, until I got to high school and heard the Tom Lehrer song and realized, "Oh. It's just math."
To the OP, I had a coworker in the 90s who made a similar argument for the Spanish Inquisition.
47: Yes. I think I'll wear a hazmat respirator and goggles.
46. This is a known line among Catholics. I had a similar conversation with a coworker about the murder of Hypatia and the subsequent canonisation of the SoB chiefly responsible (St Clement). I was informed that the advancement of the Church was in the hands of God and even if something that contributes to it looks like an atrocity to us mere mortals, it's merely because we cannot fathom the divine plan.
(Sorry if that sounds snarky, but that really was what he said, and in every other context he was one of the most sensible and rational people I ever knew.)
That kind of cod-theodicy is not limited to amateur apologists. .
45 I always wondered about this crazy "new math" that Charlie Brown complained about, until I got to high school and heard the Tom Lehrer song and realized, "Oh. It's just math."
Yeah, I always heard it was set theory, but according to Wikipedia it's also things like modular arithmetic and inequalities and logic and matrices. What's supposed to be so bad about teaching that stuff? (I could imagine some of it was being taught at an inappropriate age, I guess.)
I thought it was Cyril, not Clement, but I'm not looking.
I am trying to remember, but I think most of the objections to "new math" were along the lines of "they're dumbing down math instead of giving the children 'rithmetic, stupid liberals" rather than any coherent criticism. And yeah, I'm pretty sure it was in my math curriculum, and I remember union and intersection and little braces for sets and well of course the Antichrist.
I don't get why Christianity purged the concept of the Demiurge. It would have been such a neat answer to so many questions.
52.1. You may well be right. I'm not looking either.
53. I suspect the real answer is, "Not Greek enough."
Yeah, I guess modular arithmetic is that pinko commie anything-goes math where sometimes 9 + 4 = 1. Who cares if that's how clocks work? Burn all the clocks!
Also: is it completely standard these days for undergrads to send emails to professors asking lots of questions without ever including so much as a "hi, I'm a student who's thinking about taking your class" or any other bit of polite preamble? Just "Hey, I was wondering if you can explain..." or "Hello, Are we going to cover...?" ("I don't know! Who are 'we'? Who are you?")
51 52
... What's supposed to be so bad about teaching that stuff? ...
As I said above by trying to cover too much you can end up neglecting the most important material.
There are also issues about how well elementary school teachers understand this sort of material themselves.
55.2. The response should always be, "Who are you?" and no information.
The Children of the Puppy are a fairly unique breed, I'm given to understand.
55.2: yes. I'm tempted to get my curmudgeon on by saying that this a byproduct of so much of their communication taking place via text and IM, but I don't think that's me being curmudgeonly so much as me being right.
And yesterday I got a phone call from someone in Britain who was "in the process of arranging to get a fellowship to study" here, who wanted a detailed explanation of how my class differs from someone else's class ("They're kind of covering completely different subjects") and who never once told me his name. Today he sent me a follow-up email and what he lists as his name looks kind of fake. Plus the "From:" line reads "Universe!" followed by the-unlikely-name@gmail.com.
oh hey (maybe you just answered this) do you get emails from cranks with theories yet?
But hey, at least they're all way more polite than the other professor I have to deal with the most!
62: Yeah. Not as many as I got when I was a postdoc at Pr/nceton, though. I think they just grab the list of all researchers on the Pr/nceton and Inst/tute websites and spam them.
do you get emails from cranks with theories yet?
And, if not, would you like to?
I've now possibly manufactured a memory of reading a previous conversation on Unfogged that made me realize I'd given too much credence to some ignorant anti-New Math propaganda. It didn't stick, evidently.
How much of the right wing obsession with phonics is a direct result of the fact that they got the Hooked on Phonics people to sponsor their radio shows?
60 - Yeah, the phonics/whole language thing is amazingly fraught with political meaning among these people for reasons I don't understand.
My grandmother loves to rant about how schools no longer teach reading, or arithmetic, or real history. She heard it all on Fox or from Rush Limbaugh or whatever. And then she waxes rhapsodic about the rigorous education she got as a child in Virginia. But as far as I can tell her knowledge of American history comes entirely from reading romance novels and hagiographies of Ronald Reagan. But, since she knows what the Civil War was really about (hint: not slavery!), she had a much better education than her grandchildren.
"Through the Negro spiritual, slaves developed patience to wait on the Lord and discovered that the truest freedom is freedom from the bondage of sin."
And then the spiritual morphed into soul and R&B and we rediscovered the delights of sin.
That link in 5 is nice...good to see a modern humanist with some actual sophistication about human nature, surprisingly rare.
My grandmother loves to rant about how schools no longer teach reading, or arithmetic, or real history
At my kids school, it seems like all they teach is reading and arithmetic, to the detriment of all the other subjects kids don't get tested on.
At my kids school, it seems like all they teach is reading and arithmetic, , to the detriment of all the other subjects kids don't get tested on.
Mine, too, at least compared to what I remember of grade school.
I'm actually fine with this. Primary education should emphasize skills that can be used to build more knowledge in the future.
I'm actually quite reactionary when it comes to a lot of issues in early pedagogy. I'm pro-phonics and against teaching set theory and non-base 10 arithmetic at an early age.
It'd be quite nice if kids did other things, too. History, and geography, and art, and music, and, etc, etc. I don't think it's too early to start any of that stuff. It also makes for a more interesting school experience.
this crazy "new math" that Charlie Brown complained about
Charlie's working with an entirely different sort of crazy these days.
We did all kinds of crap. Art, music, woodworking, "social studies" (now that is a '70s concept). I bet a lot of it has been cut from those schools by now, rich town though it may be.
but I think most of the objections to "new math" were along the lines of "they're dumbing down math instead of giving the children 'rithmetic, stupid liberals" rather than any coherent criticism.
This is a lot of it. But also there is some long-standing tensions between math ed people and some pure mathematicians, and those pure mathematicians end up forming alliances with generally conservative people who are suspicious of math reforms because it seems too feel-good.
IMO the pure mathematicians who take on this fight are being obtuse assholes who are unwilling to consider that math concepts are really difficult for lots of people, and that their experience learning the basics is not universal.
OTOH, Caroline is learning a whole new algorithm for long multiplication that actually seems pretty nifty.
I'm actually fine with this. Primary education should emphasize skills that can be used to build more knowledge in the future.
Well, I'd like to see more emphasis on social skills, and more coverage of touchy-feeley topics like how to get your way without yelling at people.
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My daughter is being fragile and I keep offending her (like this) and she storms off in tears (whatever) leaving her goddamn radio of christmas carols playing in her absence. Then when she's ready to reconcile it pisses her off if I've turned off the christmas carols. But christ, who can stand them?
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Oh yeah, and philosophy and religious studies (_not_ religious indoctrination), and ... and ...
79 reminds me that this is the weekend we get to return the Justin Bieber Christmas cd to the library!
Nia's school definitely does a lot with social skills, though I suspect you might be less likely to find it at a school for a wealthier population. I have no idea how much her math-and-reading focus is normal since she started out more than a year behind in both and is probably still going to end up repeating the year, which I'm a little conflicted about to the extent that it reflects poorly on us. Since the biggest hindrance to her learning is her inability or unwillingness to shut her mouth while the teacher is talking, I think having an extra year to mature definitely wouldn't hurt. But I was also looking forward to have a year's gap between her and Mara instead of having their grades back-to-back.
Based on yesterday's classroom observation by me, my daughter's pre-K curriculum seems to consist of writing the letter "R" in many different ways until extreme boredom sets in.
But I think it's mostly about social skills actually?
My brother and sister-in-law (and by extension, my parents) talk constantly about how amazing their ivy league affiliated elementary school is, and it drives me up the fucking wall. Partly because it really probably is infinitely better than the public schools here, which appear to adhere to the endless-worksheets-for-kindergartners theory of homework assignments. Ugh.
I think new math was an attempt to popularize Bourbaki.
Reference: Lucienne Felix (1960) The Modern Aspect of Mathematics.
How would Bourbaki have felt about being popularized?
86 - We should ask his frequent collaborator Jean Dieudonné.
Ugh, don't bother talking to that guy, he's totally insufferable. Thinks he's god's gift to mathematics or something.
88: When I was in high school, one of the seniors was reading Dieudonne's real analysis book, and kept making that joke.
I adored New Math and set theory and prepositional calculus in middle school -- come to think of it, in grade school -- but my family played WffnProof when I was small, so I'm hardly evidence that one could drop New Math on everyone fruitfully.
Dang, it's been useful since, though. Recognizing a surjection out of context pretty much bought me a house.
WffnProof is the worst game ever invented.
I'm buying a house in your neighborhood, clew. I need that kind of negotiating advantage.
Gotta find your own inefficiency to arbitrage, Walt. That one got scripted away a decade ago.
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Swimming during pregnancy: awesome. I recommend it to everyone pregnant.
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I love that we now have so many pregnant commenters that a recommendation like that seems totally appropriate.
One of my nephews (in California!) was doing lots of set theory homework in third grade. Datum!
93: And when you finally get out of the pool, doesn't it first feel like you're trying to stand on some weird planet with extra gravity?
93: My wife used a tub to labor but not deliver in. She swore up and down about how much better it was the very instant she got into it.
97: You know, I didn't notice that so much as I was just happy that I'd managed a good cardio workout without my hips hurting. I did notice that it is actually pretty hard to swim with good form when one's core strength is otherwise occupied.
Oh man, I'd love to labor in a tub. There's one at the hospital, but it was broken when I had Hokey Pokey and so I don't want to get overly attached to the idea. If it's in use by someone else, for example, you're out of luck.
Could you just flood your delivery room?
100: There's one at the hospital, and it's reserved for people planning unmedicated births, and I'm told it's almost never in use.
I don't know for sure, but my guess is that 102 is exactly the case here, too. They've only had it for a couple years, and it's part of the whole "Our C-Section Rate Is Sky-High And We're Trying To Re-Brand Ourselves As A Not Entirely Hostile Place To Have Your Baby" program they undertook.
The hospital where I'm going has tubs in most of the rooms, but I'm not sure you can actually deliver in them. I think they're just for laboring.
The internet tells me that water birth is available in one of the six laboring rooms. So you can't exactly plan on it, because that one might be in use.
The hospital (one of the oldest maternity units in the world, apparently) we are using has a birth centre (for midwife led quasi-homebirths) and I think they have pools there. Afaik Mrs ttaM is more interested in the maximum drugs and un-homely birth experience, though.
A few more data points for the Aquatic Ape hypothesis.
Laboring in a tub was very helpful, not that I did it for long because Newt was an impatient child, and getting out of it to deliver wasn't much of an imposition. I figure it wouldn't be compatible with an epidural, though.
I've done one with meds and one without meds and the inescapable conclusion is that every single person and situation is so damn variable that barely anything useful can be said. For me, the time with meds was insanely more frustrating because the goddamn nurse (who was this notoriously mean person, according to the doula) dragged her feet on the epidural for several hours. Eventually when the anaesthesiologist showed up, she said "I hadn't even been told someone was waiting for an epidural until fifteen minutes ago." I found it crazy stressful to be waiting for an epidural that was never coming. (Once it arrived, it was great.)
And there are overly specific reasons why I think I'll be okay without it, this time. So, have a long helpful-to-no-one comment.
OTOH, Caroline is learning a whole new algorithm for long multiplication that actually seems pretty nifty.
Go on then, what is it?
I don't know exactly, obviously, but they tend to be methods that help kids understand what exactly is going on when you multiply. So to multiply 57 and 34, say, you break them into 50, 7, 30, and 4, and then there's some diagram to help you keep track of the moving parts as you do the 3 multiplications and add them together. Much more tied to the actual logic of multiplying than the version I learned as a kid.
There was some Japanese version I saw recently that involved colors. For the example above you'd draw a green line and label it 50, and a yellow line parallel to it and label it 7. Then transversally you'd have a green line labeled 30 and a yellow line labeled 4. So each of the intersections is an easy product to compute, and then you add them all up.
This is the method Caroline is learning. Wikipedia now informs me that it is not new at all. It can actually be found in 13th century Arab sources.
This is the method I learned in school, which I bet will be familiar to 40-somethings the world-round.
Mine were taught some thing that involved writing the two numbers to be multiplied along the top and right side of a grid of boxes. They seemed to like it, and taught it to me once or twice, but I've forgotten how it works.
But also there is some long-standing tensions between math ed people and some pure mathematicians, and those pure mathematicians end up forming alliances with generally conservative people who are suspicious of math reforms because it seems too feel-good
So was I incorrect to say that the math-ed people were confusing logical foundations with psychological mechanisms for understanding things? (Perhaps you don't even want to make that distinction.)
The usual multiplication algorithm is quadratic in time. Why don't we teach kids the Karatsuba algorithm so they can multiply in n-to-the-log-base-2-of-3 time?
essear is making good sense. Sorry, did I say "making good sense"? I meant "gradually going entertainingly insane from the course planning process".
Anyhow, mental abacus is the way to go, right?
113.2: It's never occurred to me that what we learned was a "method." It's still hard to think of it that way, as opposed to How Math Works. Anything else feels gimmicky and sort of cheat-y.
It's entirely possible I was pwned by essear because I have no idea what 116 says.
I was taught the method Rob was taught, though I've come across the lattice method. It seems to me to involve an awful lot of unnecessary fiddling around, though I suppose it's what you're used to. Also the lattice method is much harder to do in your head for comparatively small numbers like the ones in the Wikipedia example.
119: The Karatsuba algorithm is a method for multiplying numbers that is faster for gigantic numbers (like 30 digit numbers).
Is the lattice method all that much easier to teach kids who couldn't do the other one? It seems, as chris y says, kind of fiddly. Though maybe having actual lines drawn to follow the diagonals makes it easier for some kids.
Is it just me (probably) or is everything written on Wikipedia about math unusually difficult to understand and confusing.
It may well be you but if so it's me too.
Lattice method, that's it. Although they've reverted to what normal people do now that no one cares how they're doing it, which suggests that the lattice method is more of a hassle than it's worth.
The Schönhage-Strassen algorithm is even faster, and think of all the fundamental math they'd pick up along the way.
The point of the lattice method and these others is that no kid understands why standard long multiplication works, and so it's pointless to teach it except as a parlor trick in the age of Google. These other methods are more time consuming, but what you're doing actually reflects the underlying mechanics of multiplication. So, sure, long multiplication is fast-ish and works and by all means use it once you understand what exactly multiplication means, but it's not a good use of kids' time to be endlessly drilled on it as elementary school students.
I have two words for you neophytes: chisenbop.
OK, explain to me how the lattice method teaches you how multiplication works better than the traditional method.
My thought wasn't so much that the lattice method doesn't reveal how multiplication works as that you're basically doing the same thing as in standard long multiplication (multiplying each digit of the multiplicand by each digit in the multiplier), it's just that you're also drawing a little framework to put all the numbers in. But I was always super neat and able to make all my columns line up and such; I can see how drawing out the frame might make it easier for some kids.
I did the lattice method for several years in late middle school and early high school. I think it's great, as everything has it's own place, while in the usual algorithm the carrying takes place at the same spot on the page multiple times so it is very easy to make errors. I agree that it's more transparent in terms of why it works as well.
To bring things back to the OP, I'm pretty sure I learned about lattice multiplication in an A Beka christian math book. There was a one page "cool supplemental thing" at the end of each chapter and one chapter it was this method of multiplication, and I decided I liked it better.
What Blume said. It was pretty much the same, only at an angle.
In marginally related news, Sally's algebra teacher appears to have stopped functioning (not marking work, hasn't moved through the syllabus to where they should be by now, notices that he's put accidentally nonsensical problems on tests halfway through the class period and ends up telling the kids to just skip problem five. The administration is aware, and appears to be harassing him about it, but I'm not sure that's going to help.)
So I'm teaching her algebra for the Regents, and am encouraging her to collect her friends into a study group so she can drag them along after her. If they start working now, they should be okay. From my point of view it's a pleasure, but overall I'm really annoyed.
The point of the lattice method and these others is that no kid understands why standard long multiplication works, and so it's pointless to teach it except as a parlor trick in the age of Google. These other methods are more time consuming, but what you're doing actually reflects the underlying mechanics of multiplication.
I don't understand this. In long multiplication, you multiply 748 x 456 by multiplying 748 x 6, then 748 x 50, then 748 x 400. This doesn't reflect the underlying mechanics?
No, the kids don't generally get that you're multiplying by 50 in the second term, and 400 in the third term. It's sometimes explained that way, but kids don't physically write down "50" and "400" and it doesn't sink in. What sinks down is "Then you take the 5 and multiply, and when you write down the answer, you shift it to the left by one column."
I should say that I don't really know what the lattice method is, I'm just assuming I know based on what it sounds like it would do: sort of a Punnett square of multiplication, where each factor is broken down according to single, tens, hundreds, etc.
135: FWIW, that's exactly how I was taught to do multiplication, and until just now it had never occurred to me that that's what's going on.
Rob's link in 1136 has a good explanation of the lattice method.
On the minor matter of the earliest mention of the lattice method, Rob's two Wiki links can't agree. I am amused.
I was taught the long multiplication method in 113.2 as well, and I seem to think that I didn't realize what was actually being done until some teacher or other actually spelled it out:
415 x 292 = (415 x 200) + (415 x 90) + (415 x 2)
Enlightenment! Not that there was a problem coming up with the answer via the long multiplication method -- there was just no understanding.
Fundamentally all of these things rely on the distributive law, which means they're liberal propaganda.
The usual multiplication algorithm is quadratic in time. Why don't we teach kids the Karatsuba algorithm so they can multiply in n-to-the-log-base-2-of-3 time?
But what's the constant factor?
We just had a friend over for coffee. Said friend reports that her (very bright) eighth grader arrives home from school each day, works on his homework for an hour, goes to soccer practice, has a late dinner, and then returns to work on his homework, often going to sleep after she does. Her estimate is that he's doing between three and four hours/day of work and often considerably more than that. She also insists that this is the norm, which prompted me to tell my wife that we'll either home school our kids or send them to the alternative middle school (which, unlike the alternative middle school where i grew up, is not only for people coming out of rehab/people with antisocial tendencies/people who are profoundly disabled). Anyway, NCLB must be destroyed. And as much as the Obama administration's human rights/civil liberties violations are horrifying, his educational policies may well be equally bad in context.
Also, our ten-year-old's teacher is pulling a 134. I can't tell if she's having a mid-year slump, a breakdown, or if something else is amiss. In the end, it doesn't really matter; she's effectively stopped teaching the class. Many parents are up in arms, perhaps justifiably, whereas our mild concern is further muted by a sense of relief that our son's homework load has dropped from 1.5-2 hours to 0-30 minutes.
And before anybody asks, yes, I think preparing children for standardized tests is the same as or maybe even worse than waterboarding them or killing them with drones.
I don't think many people here disagree with that sentiment at all, VW, though I'm sure bob will take you to task for even the hint of the suggestion that preparing children for tests might not be worse than waterboarding them.
134
... The administration is aware, and appears to be harassing him about it, but I'm not sure that's going to help.
Good thing he has a strong union or he might lose his job or something.
which prompted me to tell my wife that we'll either home school our kids or send them to the alternative middle school
Yeah, fuck that. I just sent mine to the school where my wife teaches. No one's trying to send four hours of homework a day home with poor minority kids because they know it's not going to happen.
Do kids not have study periods -- we called them study hall -- during school hours any more? I feel as though I had at least one, maybe two, study halls per school day, at 45 minutes each, and I feel as though I got a fair amount of my homework done then. Certainly you could knock off your math homework during one.
I'm just realizing that I'm thinking of high school, though.
Study hall was an option in high school, but only the most slack students took it. There was always a more interesting/good-for-the-transcript elective to be had.
re: multiplication
I read a book years ago on calculating prodigies that had some of the mental algorithms they use in an appendix. There was a mental multiplication algorithm in there that was labelled 'cross-multiplication' although I think that has a different more common usage.
It was very mentally efficient.
We had study hall in high school, but I didn't take it or a lunch so that I could fit in extra science classes. I remember the homework being manageable, but I think I may have just had more energy back then.
148: The schools here bother me a little.
Usually "cross-multiplication" means a clearing denominators in an equation involving rational expressions. The kids are sometimes taught it in the heinously diagrammy way where they don't get that they're just multiplying both sides of the equation by the denominator.
he kids are sometimes taught it in the heinously diagrammy way where they don't get that they're just multiplying both sides of the equation by the denominator.
Maybe because the teachers don't understand it?
I remember asking my sixth grade math teacher why a number raised to the zeroth power was one. Her answer was something like: "That's just what some mathematicians decided it should be a long time ago. There's no real reason." Sigh.
Yeah, there is a fair amount of that - teachers without a deep understanding of the math they're teaching. Modern teaching training really makes a great effort to address that particular problem, but who knows how well it sticks, and who knows how well the training is delivered, and who knows how well they maintain that ethic over decades, or what kind of training they received decades ago.
Also, my math ed friend is fond of saying that you absolutely cannot force a teacher to do anything they don't want to. They will go through the motions and completely undermine you with eye-rolls and side comments, if they're not on board.
More or less true everywhere in life, I suppose.
Since I start teaching for the first time in two days, I'm encouraged by 156.
148: The schools here bother me a little.
Heh, how so? Not that I haven't noticed some things in my wifes district. Also, fuck this blizzard. I guess at least it's on a sunday with light traffic.
Yeah, there is a fair amount of that - teachers without a deep understanding of the math they're teaching.
I'm sure I've told this before, but a friend of mine specializes in teaching college students who are planning to be high school chemistry teachers. One of the examples she gives of things her students don't know (and therefore can't explain) is the difference between melting and dissolving.
160: You've told it before, but it's still horrible!
Crap, I'll give this a shot: melting is, like ... melting the same thing, but melted. When exposed to heat. Er. Dissolving is, uh, you know, dissolving one thing in another, and heat is not required, and the result is a new, third, thing.
?
I think I'm better off in the math arena. At least I was shocked by 154's "That's just what some mathematicians decided it should be a long time ago. There's no real reason."
159: Larger classes, lower standards. And fuck the entire month of January and its unholy weather.
Yeah, it still makes me shuttle wildly between pride that I - a humanities PhD! - know the difference, and despair that chemistry teachers can't figure it out with two seconds of thought.
162: Melting involves a change of state.
165: Right. I was on track to that (the application of heat being the tip-off).
My entry in the Blume challenge:
Melting is when the average kinetic energy increases to the point where the intermolecular forces are no longer able to hold them together. My impression of dissolving is that the molecules of the solid form weak bonds with the solvent lowering the energy required for escape.
162: Melting involves a change of state.
If this is supposed to suggest that sugar dissolved in water is still solid, then I confess to confusion.
Melting is what an ice cube does in water. Dissolving is what salt does in water.
Glass hand dissolving into ice petal flowers revolving.
All the years combine, they melt into a dream.
Melting involves a change of state.
"We're not in Kansas anymore, bitch!"
160: I spend a fair amount of my time teaching middle and high school teachers how to teach history. The good news is that they're usually very eager to learn.
I was about to say the problem is that "melting" is not a scientific term (this is mostly because I immediately think of cheese melting), but then I recalled that we do speak of a freezing point and a melting point.
America, dissolving pot or melting pot?
Melting is when one substance, a solid, becomes a liquid.
Dissolving is when the molecules of one substance, a solid, become dispersed among the molecules of another substance, a liquid.
Although a gas could also be dissolved in a liquid.
I don't think a liquid can be dissolved in a liquid. You would just call that a mixture.
176 --- don't we distinguish mixtures and solutions? (Or, rather, aren't solutions a subset of mixtures?)
Ice in water seems to me to be a particularly confusing example. Can you really say the ice is melting and not dissolving? Is it that the warmer water heats up the ice causing the bonds to break, or is it that the bonds in the water are pulling it off? Or more likely some combination of the two.
Colloids involve neither phase transitions nor dissolution, do they?
177: Yes, I remember learning that in high school chem. Solutions are transparent because of something or other, while mixtures contain things in different states and you can usually see the difference between the two more or less just by looking.
Also, I like the lattice method, which I've never heard of before, but I have to think about how it gets at the underlying math more than with traditional long multiplication, but I think my math education was weird in that I think I learned long multiplication and the socialist redistributive property at about the same time so everything was spelled out for us.
Colloids involve neither phase transitions nor dissolution, do they?
No, but they do involve the molecules of one substance being dispersed in the molecules of another, don't they?
I've forgotten my high-school chemistry and now I'm confused. Wikipedia tells me that a colloid is not a solution because it isn't homogeneous, but also says that in a colloid one substance is dispersed evenly throughout another. Isn't "dispersed evenly" basically the definition of "homogeneous"?
All I remember is that colloidal oatmeal soothes your skin but steel cut tastes better.
Not really I guess? Like on a sufficiently coarse level it's not really homogeneous? I dunno, seem to remember the test is more to do with filtering it (?). Also confusingly what about homogenised milk guys.
It kind of looks like there's an arbitrary numerical cutoff on the scale at which it can be inhomogeneous-- or, what I guess isn't exactly equivalent, on the size of the dissolved particle.
Is it arbitrary or is it when surface equations start being uselessly noisy?
Chemistry: stamp collecting. (Sorry guys, guys, please don't poison me some of my best friends are chemists, guys please...)
It kind of looks like there's an arbitrary numerical cutoff on the scale at which it can be inhomogeneous-- or, what I guess isn't exactly equivalent, on the size of the dissolved particle.
As Mourelatos points out, "few, if any, stuffs are homoeomerous through and through. With many the homoeomery breaks down even before we reach fine grain---e.g., fruit cake." There is certainly a reasonable sense of even dispersal on which the bits of candied fruit might be evenly dispersed throughout the cake.
I was hoping you would know.
Or no I mean like uh equations that treat interactions between the substances in the colloid as involving objects with volume and interfacing surfaces as opposed to idealized points in solution.
So what's a good way to explain why anything to the zeroth power is one?
Sounds plausible! I guess a nanometer is kind of at the "large by small-molecule chemistry standards but not, like, large large" level. Probably something different does happen there.
192: Not sure if this is a good way, but 4^3 / 4^3 equals one, and it also equals 4^(3-3) = 4^0.
I really need to find some kind of Chemistry for Dummies Physicists book, because I feel like there's actually a lot of interesting stuff there that could be worked out at at least a handwaving, order-of-magnitude kind of level but that in high school chemistry was just a bunch of poorly justified rules of thumb.
So what's a good way to explain why anything to the zeroth power is one?
Zero to the zeroth power is undefined.
Chemistry was the only AP class I actually finished in high school but I'll be damned if I can actually remember much of it.
Other than that, n0 = 1 is the only way to preserve the identity nx+1 = nxn.
197 - I mean, sort of. But really, it's 1. Now Cauchy's angry ghost will haunt me.
That's the angry ghost of Dedekind. Cauchy will just violate my personal space without ever quite touching me.
The only thing I remember from chemistry is mole.
Admittedly a more traditional ghosty procedure.
209: With a number of avocados or something.
We just had a nice little conversation about how taking something to the zeroth power leaves you with the multiplicative unit.
"Oh sure," I said, "that makes sense. If I take you to the zeroth power I am leaving you out of it altogether, and that leaves the unit. Like, if there were something like exponents for addition -- let's call it taking something to the somethingth nower, where you add it to itself that many times, if you take it to the zeroth nower, that leaves you with the additive unit, so zero."
Then Snark pointed out that this was called multiplication, and in that moment I was enlightened.
In various Lisps, (+) returns 0 and (*) returns 1. So thoughtful!
I feel like when I was in high school I must have known at least four different people who did some kind of science fair project or whatever on the idea that if multiplication is repeated addition, and exponentiation is repeated multiplication, obviously the next cool thing is repeated exponentiation, right? And then you can repeat that, and then...
It doesn't seem to lead anywhere very interesting, though.
So what's a good way to explain why anything to the zeroth power is one?
5 to the 3rd power, divided by 5 to the 2nd power, is 5 to the 1st power, or 5.
5 to the 3rd power, divided by 5 to the 3rd power, is 5 to the 0th power, or 1.
If zero to the zeroth means anything, then surely it should be the number of maps from the empty set to the empty set, which is 1.
213 - Leads to some amusing notation, though.
Why is that so obviously the what it should mean?
It doesn't seem to lead anywhere very interesting, though.
(a) how is that science fair project and
(b) It leads to fuckin' huge numbers!
I feel like 215 would have really appealed to me a decade ago.
I can't but read 219 as a version of "I can see why you think that—I used to think it myself".
217a: Well, who'd go to a fair without bringing their handmaiden?
It's like you skipped the Star Wars prequels, essear.
Or their queen? How can it be both?
So your friends thought iterated exponentiation was metaphysics?
What does winning at the metaphysics fair really mean, anyhow?
Isn't there some theorem that says, and I'm speaking roughly, "Don't be silly; exponentiation is as far as you need to go to do anything interesting."?
Please tell me 225 is a Severn Darden reference.
What about defining a total computable function which is not primitive recursive?
Hey, remember that PBS documentary about Fermat's last theorem that said something like "the basic mathematical operations are addition, subtraction, multiplication, division ... and modular forms"? And then had some kind of trippy animation of a glowing spinning donut or something? What was up with that?
Good grief that is possibly the most impenetrable comment I have made. I apologise to all.
My entry for the eighth grade metaphysics fair was an object that continued existing even when no one was experiencing it, but I had a devil of a time convincing the judges that what they experienced after a period of not experiencing it was numerically and not just qualitatively identical to what they had experienced at first.
I wonder whatever happened to that thing.
Modular forms are pretty cool. I want to know more about them. I wonder if anyone would notice if I lectured about those instead of what I'm supposed to be teaching this week.
229: Hey, I remember that! Not that I have any real idea what it was about. I vividly remember the spinning donut thing, though.
It's like you skipped the Star Wars prequels, essear.
Good grief that is possibly the most impenetrable comment I have made.
Is no one around here willing to work hard for penetration these days?
That was probably not the most clever response to that prompt imaginable, but no one else was stepping up.
There is never a situation in which a comment has to be penetrated, teo, even if apo tells you otherwise.
High penetration can be problematic for system reliability. It's often best to stick to low penetration unless you can be sure that the existing system is robust enough to handle potentially large but unpredictable injections of juice.
228: Is that interesting? Don't be silly.
The lattice method of multiplication is close to the box method I used to teach multiplication of polynomials, so I expect that students who learned to multiply with the lattice method might pick up polynomial multiplication quicker when it comes to algebra. (The box method I used is basically the lattice method without splitting the individual products into tens and ones, and adding the diagonals up without the carries. Just make sure to include any missing terms with zero coefficients when setting up the product.)
Agree with most of the explanations for why x^0 is 1. Most of the rules for exponentiation are explained by starting with the rules for exponentiation by positive integers (which are pretty intuitive in terms of multiplication and division), and then saying: we want to define expoonentiation by {0, negative numbers, fractions, real numbers} in such a way that these rules remain valid when we extend the exponents to include this new set of objects.
It doesn't seem to lead anywhere very interesting, though.
Repeated exponentiation leads towards the top right hand corner of the page, with the typeface getting progressively smaller. Whether it's interesting depends what you find in that corner, in really tiny type.
The On-Line Encyclopedia of Integer Sequences* has an interesting discussion on 00. Among other points they mention that The binomial expansion formula requires 0! = 1 and 00= 1.
And not conclusive, but for fans of continuity (or graphical visualization) the limit of x0 ->0 is compelling. Upon initially hearing it, I think most people intuitively think of the limit of 0x -> 0 (or at least for x being positive integers) which is 0 for all positive values of x, but infinity for all negative values of x (and of course one is halfway between those two--multiplicatively speaking, which we are).
*You're learning about it mere minutes after I did.
*You're learning about it mere minutes after I did.
What if I first read your comment three hours from now?
247
The limit of x**y as x,y -> 0 is usually 1 but when you approach (0,0) along the x axis it isn't. So you can't make x**y a continuous function at (0,0) but it mostly makes sense to define 0**0 as 1.
247: I think you should have let someone reading it three hours from now expose that flaw in my logic. But nooooo, you had to hog all the glory. Well done.
If you're only talking about exponentiation of non-negative integers (0, 1, 2, 3, ...) then 0 to the 0 is definitely 1. Exponentiation of real numbers is actually very confusion, for example, you can't raise negative real numbers to most powers sensibly. If you're just looking at raising non-negative real numbers to non-negative real powers, then you have the problem that it's discontinuous at 0 no matter what definition you use (as Shearer says).
It all makes a bit more sense when you look at complex numbers, where you see that a to the b is exp(b log a) and log is not defined at 0 and is multivalued elsewhere (e.g. log 1 can be any multiple of 2 pi i). So from that point of view it makes most sense to say 0 to the 0 is undefined.
Speaking of special purposes, something from someone who some* people on this blog will recognize appears to be out. Learned by me via Josh Marshall shout out.
*Actually, probably all people.
Exponentiation of real numbers is actually very confusion, for example, you can't raise negative real numbers to most powers sensibly.
I was not aware of this. Can you explain, please.
What's (-1)**(1/2)? Whatever it is (i or -i) it's certainly not real. (-1)**(1/3)? Now there's 3 choices. What about (-1)**(2/3)? What consistency do you want between your choice for (-1)**(1/3) and (-1)**(2/3)? Note you need to fix all of these rational powers before you even have a hope of defining say (-1)**pi as a limit.
But those ambiguities apply equal well to roots of positive numbers, no? I guess the distinction you're going for is that choosing a convention and sticking to it works in that case?
Right. For a**b with a positive real and b real, there's a nice well-defined way to do things that stays entirely inside the reals and has nothing confusing happen. Once you introduce a being negative then there's no way to stay inside the reals, and once you go to the complex numbers you introduce all the usual issues (picking a branch cut, being careful because the obvious identities don't make sense when you jump over the branch cut, etc.).
There's 3 different ways to talk about exponentiation that make sense in their own worlds:
a and b are non-negative integers, and a**b is the number of maps from an a element set to a b element set, this stays inside the non-negative integers.
a is a positive real number and b is a real number. Answers stay within positive real numbers.
a and b are complex numbers and a**b = exp(b log a) where log is multivalued so you have to deal with the usual issues that gives.
Makes sense.
In my corner of the world sometimes branch cuts are good and friendly and rational functions are mysterious and scary.
Yeah, what I was saying in 244 is mostly focused on the second case of 256, where the base a is a positive real. But you need to understand the definitions for exponentiation over real exponents to make sense of exp and log as continuous functions, and then understand the extension of exp to the complex numbers via Euler's formula, before the definition in 256:last makes much sense.
Understanding the rules for {negative, rational, real} exponents as logical extensions of the intuitive rules for positive integer exponents as opposed to arbitrary additional rules to be memorized was often a huge advance for a lot of my algebra students when I was tutoring.
You guys aren't into Riemann surfaces?
I wouldn't date a Riemann surface, if that's what you're asking.
But you wouldn't turn down a little interlude between the sheets, right?
Well boy my attempt to keep people talking about what they were talking about sure was not a successful attempt to do that.
258.last is very true. I always spend a decent bit of time on that in Calculus class. But 0**0 isn't really determined just by those intuitive rules.
I bow to Sifu.
I think I still don't understand real exponents.
If a is a fixed positive real, then the function a**x from the rational numbers to the positive real numbers is continuous. Since the rationals are dense in the reals, it extends to a unique function a**x from the real numbers to the positive real numbers. That is to say, take your number x and write it as a limit of rationals x_i. Define a**x as the limit of the a**x_i.
My best friend, he shoots water rats and feeds them to his geese
Don't you think there's a place for you in between the sheets?
267: What PETGI said. If you want to raise a**pi, you can look at a**3, a**3.1 = a**(31/10), a**3.14=a**(314/1000, etc., keeping in mind that a**(31/10) is the tenth root of a**31, and so on. Showing that this sequence converges to a limit left as an exercise for the reader... although it might help to note that, say, a**(314/100) has to be between a**(31/10) and a**(32/10). Anyhow, the (real) limit is what we define a**pi to be.
Oh, if that's all it is, fine. I thought there was going to be something really head-spinning.